Author
Valentin Bahier
Bio: Valentin Bahier is an academic researcher from Institut de Mathématiques de Toulouse. The author has contributed to research in topics: Unit circle & Permutation matrix. The author has an hindex of 3, co-authored 6 publications receiving 16 citations.
Papers
More filters
••
TL;DR: In this paper, the authors considered the ensemble of permutation matrices following Ewens' distribution of a given parameter and its modification where entries equal to 1 in the matrices are replaced by independent random variables uniformly distributed on the unit circle.
Abstract: We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens’ distribution of a given parameter $$\theta >0$$
and its modification where entries equal to 1 in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the elements of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a single arc, we extend this result to the case where the length goes to zero sufficiently slowly when the size of the matrix goes to infinity. Finally, we investigate the behavior of the largest and smallest spacings between two distinct consecutive eigenvalues.
5 citations
••
TL;DR: In this article, the authors consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle.
Abstract: We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if $a$ and $a+b$ denote the endpoints of $J$, we still have an asymptotic normality for the number of points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$] and [$a,b \to \infty$ with $b$ proportional to $a$].
4 citations
••
TL;DR: In this paper, the authors studied the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures, which are one-parameter deformations of the uniform distribution on the permutation group.
3 citations
•
TL;DR: In this article, the authors consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle.
Abstract: We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if $a$ and $a+b$ denote the endpoints of $J$, we still have an asymptotic normality for the number of points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$] and [$a,b \to \infty$ with $b$ proportional to $a$].
3 citations
•
TL;DR: In this article, the authors studied the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures that are one-parameter deformations of the uniform distribution on the permutation group.
Abstract: We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation group We also look at some modifications of permutation matrices where the entries equal to one are replaced by iid uniform variables on the unit circle Once appropriately normalized and scaled, we show that the characteristic polynomial converges in distribution on every compact subset of $\mathbb{C}$ to an explicit limiting entire function, when the size of the matrices goes to infinity Our findings can be related to results by Chhaibi, Najnudel and Nikeghbali on the limiting characteristic polynomial of the Circular Unitary Ensemble
2 citations
Cited by
More filters
••
TL;DR: In this paper, a logarithmic correlation structure for the maximum modulus of a uniform permutation matrix on the unit circle was shown to be a random field on the circle.
Abstract: Let $P_N$ be a uniform random $N\times N$ permutation matrix and let $\chi_N(z)=\det(zI_N- P_N)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically, \[ \sup_{|z|=1}|\chi_N(z)|= N^{x_0 + o(1)} \] with probability tending to one as $N\to \infty$, for a numerical constant $x_0\approx 0.652$. The main idea of the proof is to uncover a logarithmic correlation structure for the distribution of (the logarithm of) $\chi_N$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. Unlike the well-studied \emph{CUE field} in which $P_N$ is replaced with a Haar unitary, the distribution of $\chi_N(e^{2\pi it})$ is sensitive to Diophantine properties of the point $t$. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory.
10 citations
•
TL;DR: In this paper, it was shown that the normalized count of the number of eigenangles in a fixed interval of a permutation representation evaluated at a random element of the symmetric group σ ∈ \mathfrak{S}_n$ converges weakly to a compactly supported distribution.
Abstract: Let the term $k$-representation refer to the permutation representations of the symmetric group $\mathfrak{S}_n$ on $k$-tuples and $k$-subsets as well as the $S^{(n-k,1^k)}$ irreducible representation of $\mathfrak{S}_n$. Endow $\mathfrak{S}_n$ with the Ewens distribution and let $\alpha$ and $\beta$ be linearly independent irrational numbers over $\mathbb{Q}$. Then for fixed $k > 1$ we show that as $n \to \infty$, the normalized count of the number of eigenangles in a fixed interval $(\alpha, \beta)$ of a $k$-representation evaluated at a random element $\sigma \in \mathfrak{S}_n$ converges weakly to a compactly supported distribution. In particular, we compute the limiting moments and moreover provide an explicit formula for the limiting density when $k = 2$ and the Ewens parameter $\theta = 1$ (uniform probability measure). This is in contrast to the $k = 1$ case where it has been shown previously that the distribution is asymptotically Gaussian.
5 citations
••
TL;DR: In this article, the authors consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle.
Abstract: We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if $a$ and $a+b$ denote the endpoints of $J$, we still have an asymptotic normality for the number of points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$] and [$a,b \to \infty$ with $b$ proportional to $a$].
4 citations
•
TL;DR: In this article, the authors consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle.
Abstract: We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if $a$ and $a+b$ denote the endpoints of $J$, we still have an asymptotic normality for the number of points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$] and [$a,b \to \infty$ with $b$ proportional to $a$].
3 citations
•
TL;DR: In this article, the eigenvalue point process of the permutation representation of the symmetric group (S) with the Ewens distribution was studied in the microscopic regime.
Abstract: Equip the symmetric group $\mathfrak{S}_n$ with the Ewens distribution. We study the eigenvalue point process of the permutation representation of $\mathfrak{S}_n$ on $k$-tuples of distinct integers chosen from the set $\{1,2,...,n\}$. Taking $n \to \infty$, we find the limiting point process in the microscopic regime, i.e. when the eigenvalue point process is viewed at the scale of the mean eigenvalue spacing. A formula for the limiting eigenvalue gap probability in an interval is also given. In certain cases, a power series representation exists and a combinatorial procedure is given for computing the coefficients.
2 citations